{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Maroon Emphasis" -1 256 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 257 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 258 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 259 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Blue Emphasis" -1 260 "Time s" 1 12 0 0 255 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Green Emphasis" -1 261 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis " -1 262 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 267 "System" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 60 "Examples involving the numerical \+ solution of 2nd order DE's " }}{PARA 3 "" 0 "" {TEXT 266 25 " . . elec trical examples " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanai mo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 24.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "load " }{TEXT 0 7 "desolve" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " } {TEXT 262 7 "DEsol.m" }{TEXT -1 32 " is required by this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 121 "It can be read into a Maple session by a command similar to the one that follows, where the file path gives \+ its location." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "read \"K:\\ \\Maple/procdrs/DEsol.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 45 "load root-finding procedures etc. includi ng: " }{TEXT 0 31 "secant,allroots,findmax,findmin" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 262 7 "roots.m" }{TEXT -1 32 " \+ is required by this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 121 "It ca n be read into a Maple session by a command similar to the one that fo llows, where the file path gives its location." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "read \"K:\\\\Maple/procdrs/roots.m\";" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 14 "An RLC c ircuit" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 13 "" 1 "" {TEXT -1 1 " " }{GLPLOT2D 272 178 178 {PLOTDATA 2 "6+-%%TEXTG6$7$ $\"\"(\"\"!$\"\"%F)%\"LG-F$6$7$$\"$v#!\"\"$!#lF2%\"CG-F$6$7$F'$!#BF)% \"RG-F$6$7$$!$D\"F2$!#5F)%%V(t)G-%'CURVESG6#717$$!\"&F2!#?7$F)$!$&=F27 $\"\"\"$!$:#F27$\"\"#FM7$\"\"$FQ7$F+FM7$\"\"&FQ7$\"\"'FM7$F(FQ7$\"\")F M7$\"\"*FQ7$\"#5FM7$\"#6FQ7$\"#7FM7$$\"$D\"F2FK-FE6,7$7$$\"%a7!\"#F)7$ \"#CF)7$7$F?F)7$F2F)7$F[p7$F\\p$!#%*F27$7$F\\p$!$1\"F27$F\\pFK7$7$\"#? 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+Pt0,(*FB7$$\"+C#[%y**Fcr$!+\\!=)*)**FB7$$\"+]&[d%)*Fcr$!+TiJ!))*FB7$$ \"+)R)>>(*Fcr$!+.u\"pP*FB7$$\"+*p+lg*Fcr$!+f/p*\\)FB7$$\"+lh![^*Fcr$!+ ywg$G(FB7$$\"+Y\"G/X*Fcr$!+W/:xdFB7$$\"+s4f=%*Fcr$!+`kPSSFB7$$\"+*p@KU *Fcr$!+.a_U@FB7$$\"+&\\$zm%*Fcr$!+gie#f\"Fgal7$$\"+oqA]&*Fcr$\"+!Hd.$= FB7$$\"+'*R&Gn*Fcr$\"+OE+ZPFB7$$\"+5LWK)*Fcr$\"+7oE9bFB7$$\"+Z!HD+\"F^ s$\"+sXphqFB7$$\"+2ljC5F^s$\"+`[fF$)FB7$$\"+N6&*[5F^s$\"+2-]h#*FB7$$\" +h$pZ2\"F^s$\"+u(yh#)*FB7$$\"+vwK,6F^s$\"+,'=\"****FB7$$\"+fI$y7\"F^s$ \"+C^Ut(*FB7$$\"+/Y\\`6F^s$\"+Hg4e\"*FB7$$\"+[]bx6F^s$\"+XDmx\")FB7$$ \"+`4K*>\"F^s$\"+i9@roFB7$$\"+g.>=7F^s$\"+ho#3H&FB7$$\"+(ywOB\"F^s$\"+ !p8&*\\$FB7$$\"+:'GaC\"F^s$\"+]foo:FB7$$\"+;JC`7F^s$!+sM!oC%Fgal-%*AXE SSTYLEG6#%%NONEG" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}{PARA 0 "" 0 "" {TEXT -1 64 "Given a standard series \+ RLC circuit, driven by a voltage source " }{XPPEDIT 18 0 "V(t)" "6#-% \"VG6#%\"tG" }{TEXT -1 36 " volts, the charge on the capacitor " } {TEXT 279 1 "q" }{TEXT -1 47 " coulombs satisfies the differential equ ation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L;" "6#%\" LG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*q/(d*t^2)+R;" "6#,&*(%\"dG\"\" #%\"qG\"\"\"*&F%F(*$%\"tGF&F(!\"\"F(%\"RGF(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dq/dt+q/C = V(t);" "6#/,&*&%#dqG\"\"\"%#dtG!\"\"F'*&%\" qGF'%\"CGF)F'-%\"VG6#%\"tG" }{TEXT -1 14 " ------- (i), " }}{PARA 0 " " 0 "" {TEXT -1 40 "where the resistance of the resistor is " }{TEXT 284 1 "R" }{TEXT -1 34 " ohms, the coil has inductance of " }{TEXT 263 1 "L" }{TEXT -1 48 " henrys and the capacitance of the capacitor i s " }{TEXT 264 1 "C" }{TEXT -1 8 " farads." }}{PARA 0 "" 0 "" {TEXT -1 51 "This DE is obtained by applying Ohm's law to give: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "V(t) = L;" "6#/-%\"VG6#%\"t G%\"LG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dI/dt+R*I+q/C;" "6#,(*&%#dIG\" \"\"%#dtG!\"\"F&*&%\"RGF&%\"IGF&F&*&%\"qGF&%\"CGF(F&" }{TEXT -1 2 ", \+ " }}{PARA 0 "" 0 "" {TEXT -1 19 "where, if the time " }{TEXT 265 1 "t " }{TEXT -1 25 " is measured in seconds, " }{XPPEDIT 18 0 "I = dq/dt; " "6#/%\"IG*&%#dqG\"\"\"%#dtG!\"\"" }{TEXT -1 49 " is the current in t he circuit measured in amps. " }}{PARA 0 "" 0 "" {TEXT -1 95 "The equa tion (i) is a (non-homogenous) linear differential equation with const ant coefficients." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 10 "Example 1 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L=1" "6#/%\" LG\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "R=1/4" "6#/%\"RG*&\"\"\"F& \"\"%!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "C=1" "6#/%\"CG\"\"\"" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "V(t)=2*exp(-t/5)" "6#/-%\"VG6#%\"tG *&\"\"#\"\"\"-%$expG6#,$*&F'F*\"\"&!\"\"F1F*" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 20 "Initial conditions: " }{XPPEDIT 18 0 "q(0 ) = 0;" "6#/-%\"qG6#\"\"!F'" }{TEXT -1 4 ", q'" }{XPPEDIT 18 0 "``(0)= 0" "6#/-%!G6#\"\"!F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 39 " The differential equation (i) becomes: " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*q/(d*t^2)+1/4;" "6#,&*(%\"dG\"\"#%\"qG\"\" \"*&F%F(*$%\"tGF&F(!\"\"F(*&F(F(\"\"%F,F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dq/dt+q = 2*exp(-t/5);" "6#/,&*&%#dqG\"\"\"%#dtG!\"\"F'%\"qGF'*& \"\"#F'-%$expG6#,$*&%\"tGF'\"\"&F)F)F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 46 "We can find an analytical solution as follows." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "V := t -> 2*exp(-t/5);\nL := 1; R := 1/4; C := 1;\nde := L*diff(q (t),t$2)+R*diff(q(t),t)+q(t)/C=V(t);\nic := q(0)=0, D(q)(0)=0;\ndsolve (\{de,ic\},q(t)):\nQ := unapply(rhs(combine(%)),t);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"VGf*6#%\"tG6\"6$%)operatorG%&arrowGF(,$*&\"\"#\" \"\"-%$expG6#,$*&#F/\"\"&F/9$F/!\"\"F/F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG# \"\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"qG6#%\"tG-%\"$G6$F-\" \"#\"\"\"*&#F2\"\"%F2-F(6$F*F-F2F2F*F2,$*&F1F2-%$expG6#,$*&\"\"&!\"\"F -F2F@F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"qG6#\"\"!F*/ --%\"DG6#F(F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"QGf*6#%\"tG6\"6 $%)operatorG%&arrowGF(,(*&#\"#S\"$$p\"\"\"*(-%$expG6#,$*&#F1\"\")F19$F 1!\"\"F1-%$sinG6#,$*&#\"\"$F9F1*&\"\"(#F1\"\"#F:F1F1F1F1FDFEF1F1*&#\"$ +#\"#**F1*&F3F1-%$cosGF>F1F1F;*&#FIFJF1-F46#,$*&#F1\"\"&F1F:F1F;F1F1F( F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 " It is convenient to plot the graphs of the input voltage " }{XPPEDIT 18 0 "V(t)" "6#-%\"VG6#%\"tG" }{TEXT -1 27 " and the associated charge " }{XPPEDIT 18 0 "q(t)" "6#-%\"qG6#%\"tG" }{TEXT -1 13 " and current \+ " }{XPPEDIT 18 0 "i(t) = ``;" "6#/-%\"iG6#%\"tG%!G" }{TEXT -1 2 "q'" } {XPPEDIT 18 0 "``(t)" "6#-%!G6#%\"tG" }{TEXT -1 95 " in the same pictu re even though different units are involved for each of the three quan tities." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "plot([V(t),Q(t),D(Q)(t)],t=0..15,\n color=[red,C OLOR(RGB,.9,.7,0),COLOR(RGB,.5,0,1)],\n labels=[`t`,``],legend=[\" input voltage\",\"charge on capacitor\",\"current\"]);" }}{PARA 13 "" 1 "" {GLPLOT2D 578 267 267 {PLOTDATA 2 "6'-%'CURVESG6%7S7$$\"\"!F)$\" 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"cha rge on capacitor" "current" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "A continuous numerical solution may also be ob tained using an adaptive combined order 7 and 8 Runge-Kutta method. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 208 "V := t -> 2*exp(-t/5):\nL := 1: R := 1/4: C := 1:\nde := L*diff (q(t),t$2)+R*diff(q(t),t)+q(t)/C=V(t);\nic := q(0)=0, D(q)(0)=0;\n(Qn, In) := desolve(\{de,ic\},q(t),t=0..15,type=numeric,method=rk78,output= proc_pair);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\" qG6#%\"tG-%\"$G6$F-\"\"#\"\"\"*&#F2\"\"%F2-F(6$F*F-F2F2F*F2,$*&F1F2-%$ expG6#,$*&\"\"&!\"\"F-F2F@F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#i cG6$/-%\"qG6#\"\"!F*/--%\"DG6#F(F)F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "plot([V(t),'Qn'(t),'In' (t)],t=0..15,\n color=[red,COLOR(RGB,.9,.7,0),COLOR(RGB,.5,0,1)], \n labels=[`t`,``],legend=[\"input voltage\",\"charge on capacitor \",\"current\"]);" }}{PARA 13 "" 1 "" {GLPLOT2D 473 229 229 {PLOTDATA 2 "6'-%'CURVESG6%7S7$$\"\"!F)$\"\"#F)7$$\"3')*****\\7t&pK!#=$\"3=J7z;9 St=!#<7$$\"3$****\\(=7T9hF/$\"3]*yfyc'yp2>nf25F27$$\"3;++]([D9v$F2$\"3J-2EG!QYW*F/7$$\"3c****\\iNGw SF2$\"3_h1%z@90&))F/7$$\"37++]7XM*Q%F2$\"3I@kATfN8$)F/7$$\"3/+](o%QjtY F2$\"3/Y+PLH'Q&yF/7$$\"32++]i8o6]F2$\"3&Qx[[b>/M(F/7$$\"3i******\\>0)H &F2$\"3`/1a&Q;=$pF/7$$\"3Y**\\(=-p6j&F2$\"3s*3TX8V]['F/7$$\"3d*****\\2 Mg#fF2$\"3Du!=G1fO6'F/7$$\"35+](=xZ&\\iF2$\"3))G,vAUhIdF/7$$\"3;+]i:$4 wb'F2$\"3AN^&)*y%>)Q&F/7$$\"3-++v=#R!zoF2$\"3Oil'zY3F0&F/7$$\"3q+]P4A@ urF2$\"3=*p8'zw0jZF/7$$\"3I++Dchf#\\(F2$\"3imEF\"=;#pWF/7$$\"3))**\\(o f2L#yF2$\"3a\"fqy%f<$=%F/7$$\"3M**\\7yG>6\")F2$\"3Q;uLnO7\\RF/7$$\"3w+ +voo6A%)F2$\"3C4VQ)*y-6PF/7$$\"3q*****\\xJLu)F2$\"3#[eeh,<,[$F/7$$\"3W ++v$*ydd!*F2$\"3+V]hye7oKF/7$$\"3#***\\(=1\"Ffu$\"3c+a?Y )y5R#F/7$$\"3=++vQ(zS4\"Ffu$\"3d-%e[cGT>)>F/7$$\"3!****\\73\"o'=\"F fu$\"33;+2L\"RL'=F/7$$\"3-+voz;)*=7Ffu$\"3K_Xm`4xYG\"Ffu$\"3_[prCh@S:F/7$$\"34+v=(4bMJ\" Ffu$\"3wHs)=IIgW\"F/7$$\"3;++]xlWU8Ffu$\"3o@Tep'pXO\"F/7$$\"39+]i&3ucP \"Ffu$\"3u)*>XrY$oF\"F/7$$\"3\"******\\;$R09Ffu$\"3vzH<+M:.7F/7$$\"38+ v=-*zqV\"Ffu$\"3\"3j;.?r#H6F/7$$\"33+D\"G:3uY\"Ffu$\"3MGGM+'4G1\"F/7$$ \"#:F)$\"3!*)ysNn8u&**!#>-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%'LEGEND G6#Q.input~voltage6\"-F$6%7aq7$F(F(7$$\"+0k'p3%!#6$\"+xP*)f;!#77$$\"+5 G$R<)F]\\l$\"+>kG'f'F`\\l7$$\"+A**3E7!#5$\"+AO3u9F]\\l7$$\"+ilyM;Fi\\l $\"+\"Ry?g#F]\\l7$$\"+W)z@X#Fi\\l$\"+HQZndF]\\l7$$\"+DJdpKFi\\l$\"+84% *35Fi\\l7$$\"+s@*>p%Fi\\l$\"+0_(\\,#Fi\\l7$$\"+>7T9hFi\\l$\"+d')>2LFi \\l7$$\"+0-29xFi\\l$\"+KVEX]Fi\\l7$$\"+)=HPJ*Fi\\l$\"+$**Rz,(Fi\\l7$$ \"+v\"*R#4\"!\"*$\"+\")[nn\"*Fi\\l7$$\"+JaU`7Fb_l$\"+gSxS6Fb_l7$$\"+ej o89Fb_l$\"+8\">aO\"Fb_l7$$\"+%GZRd\"Fb_l$\"+c&3[e\"Fb_l7$$\"+y'HDs\"Fb _l$\"+>yx\"Fb_l7$$\"+s?6r=Fb_l$\"+P?.c>Fb_l7$$\"+M0'\\-#Fb_l$\"+$[8/ 7#Fb_l7$$\"+(**3)y@Fb_l$\"+'em.E#Fb_l7$$\"+(H>zL#Fb_l$\"+z6&fP#Fb_l7$$ \"+(fHq\\#Fb_l$\"+$R;#fCFb_l7$$\"+7'Hjd#Fb_l$\"+$**e!)[#Fb_l7$$\"+G'Hc l#Fb_l$\"+4jG3DFb_l7$$\"+O'z_p#Fb_l$\"+PR9:DFb_l7$$\"+W'H\\t#Fb_l$\"+! 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" }}{PARA 0 "" 0 " " {TEXT -1 20 "Initial conditions: " }{XPPEDIT 18 0 "q(0) = 0;" "6#/-% \"qG6#\"\"!F'" }{TEXT -1 4 ", q'" }{XPPEDIT 18 0 "``(0)=0" "6#/-%!G6# \"\"!F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 39 "The different ial equation (i) becomes: " }}{PARA 256 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "d^2*q/(d*t^2)+1/4;" "6#,&*(%\"dG\"\"#%\"qG\"\"\"*&F%F(* $%\"tGF&F(!\"\"F(*&F(F(\"\"%F,F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dq/d t+q = 2*t/(t^2+1);" "6#/,&*&%#dqG\"\"\"%#dtG!\"\"F'%\"qGF'*(\"\"#F'%\" tGF',&*$F-F,F'F'F'F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 109 "A continuous numerical solution may \+ be obtained using an adaptive combined order 7 and 8 Runge-Kutta metho d. 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" }}{PARA 0 "" 0 "" {TEXT -1 20 "Initial conditions: " } {XPPEDIT 18 0 "q(0) = 0;" "6#/-%\"qG6#\"\"!F'" }{TEXT -1 4 ", q'" } {XPPEDIT 18 0 "``(0)=0" "6#/-%!G6#\"\"!F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 39 "The differential equation (i) becomes: " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*q/(d*t^2)+1/4;" "6#,& *(%\"dG\"\"#%\"qG\"\"\"*&F%F(*$%\"tGF&F(!\"\"F(*&F(F(\"\"%F,F(" } {TEXT -1 1 " " }{XPPEDIT 18 0 "dq/dt+q = sin*t/(1+sin^2*t);" "6#/,&*&% #dqG\"\"\"%#dtG!\"\"F'%\"qGF'*(%$sinGF'%\"tGF',&F'F'*&F,\"\"#F-F'F'F) " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "We can find an analytical solution using " }{TEXT 0 6 "ds olve" }{TEXT -1 12 " as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 187 "V := t -> sin(t)/(1+sin(t)^ 2);\nL := 1; R := 1/4; C := 1;\nde := L*diff(q(t),t$2)+R*diff(q(t),t)+ q(t)/C=V(t);\nic := q(0)=0, D(q)(0)=0;\ndsolve(\{de,ic\},q(t)):\nQ := \+ unapply(rhs(combine(%)),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VGf *6#%\"tG6\"6$%)operatorG%&arrowGF(*&-%$sinG6#9$\"\"\",&F1F1*$)F-\"\"#F 1F1!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG#\"\"\"\"\"%" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"CG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#de G/,(-%%diffG6$-%\"qG6#%\"tG-%\"$G6$F-\"\"#\"\"\"*&#F2\"\"%F2-F(6$F*F-F 2F2F*F2*&-%$sinGF,F2,&F2F2*$)F9F1F2F2!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"qG6#\"\"!F*/--%\"DG6#F(F)F*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"QGf*6#%\"tG6\"6$%)operatorG%&arrowGF(-%$IntG 6$*&,&**\"\")\"\"\"-%$cosG6#,(%$_z1G!\"\"*&#\"\"$F2F3*&\"\"(#F3\"\"#9$ F3F3F3**FF?F8F3F9F3F>F?-%$expG6#,&*&F2F9F8F3F3*&#F3F2F3FAF3F9 F3F9**F2F3-F56#,(F8F3*&F;F3F=F3F3**FF?F8F3F9F3F>F?FCF3F3F3,& \"#jF9*&\"#@F3-F56#,$*&F@F3F8F3F3F3F3F9/F8;\"\"!FAF(F(F(" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "It is convenient t o plot the graphs of the input voltage " }{XPPEDIT 18 0 "V(t)" "6#-%\" VG6#%\"tG" }{TEXT -1 27 " and the associated charge " }{XPPEDIT 18 0 " q(t)" "6#-%\"qG6#%\"tG" }{TEXT -1 13 " and current " }{XPPEDIT 18 0 "i (t) = ``;" "6#/-%\"iG6#%\"tG%!G" }{TEXT -1 2 "q'" }{XPPEDIT 18 0 "``(t )" "6#-%!G6#%\"tG" }{TEXT -1 95 " in the same picture even though diff erent units are involved for each of the three quantities." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "plo t([V(t),Q(t),D(Q)(t)],t=0..20,\n color=[red,COLOR(RGB,.9,.7,0),COL OR(RGB,.5,0,1)],\n 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{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 163 "The values given by nu merical procedures for the both the charge and the current at the end \+ of the interval agree well with those given by the analytical solution ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "tt := 20;\nQn(tt);\nevalf(evalf[14](Q(tt)));\nIn(tt); \nevalf(evalf[14](D(Q)(tt)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#tt G\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+r$3\"G#)!#5" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$!+r$3\"G#)!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$\"+Rbe4?!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Rbe4?!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "An associ ated " }{TEXT 262 11 "phase curve" }{TEXT -1 4 " or " }{TEXT 262 10 "t rajectory" }{TEXT -1 19 " may also be drawn." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "plot(['Qn'(t),'In' (t),t=0..15],color=COLOR(RGB,.5,0,1),labels=[`charge`,`current`]);" }} {PARA 13 "" 1 "" 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" }}{PARA 0 "" 0 "" {TEXT -1 20 "Initia l conditions: " }{XPPEDIT 18 0 "q(0) = 0;" "6#/-%\"qG6#\"\"!F'" } {TEXT -1 4 ", q'" }{XPPEDIT 18 0 "``(0)=0" "6#/-%!G6#\"\"!F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 39 "The differential equation (i) becomes: " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*q /(d*t^2)+1/4;" "6#,&*(%\"dG\"\"#%\"qG\"\"\"*&F%F(*$%\"tGF&F(!\"\"F(*&F (F(\"\"%F,F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dq/dt+q = sin^32*t;" "6# /,&*&%#dqG\"\"\"%#dtG!\"\"F'%\"qGF'*&%$sinG\"#K%\"tGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 109 "A c ontinuous numerical solution may be obtained using an adaptive combine d order 7 and 8 Runge-Kutta method. 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qGF'-%\"FG6#%\"tG" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 56 "wher e the current in the underlying circuit is given by " }{XPPEDIT 18 0 " i = dq/dt;" "6#/%\"iG*&%#dqG\"\"\"%#dtG!\"\"" }{TEXT -1 49 ", the inst antaneous rate of change of the charge " }{TEXT 275 1 "q" }{TEXT -1 20 " in the vacuum tube." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Example 1: " }{XPPEDIT 18 0 "mu=1" "6#/%#muG\"\"\"" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 " " {TEXT -1 5 "When " }{XPPEDIT 18 0 "mu=1" "6#/%#muG\"\"\"" }{TEXT -1 32 ", the differential equation is: " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*q/(d*t^2)+``(q^2-1);" "6#,&*(%\"dG\"\"#%\"qG \"\"\"*&F%F(*$%\"tGF&F(!\"\"F(-%!G6#,&*$F'F&F(F(F,F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dq/dt+q = 0;" "6#/,&*&%#dqG\"\"\"%#dtG!\"\"F'%\"qGF'\" \"!" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 28 "with the initial c onditions " }{XPPEDIT 18 0 "q(0)=1" "6#/-%\"qG6#\"\"!\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "`q '`(0) = 0" "6#/-%$q~'G6#\"\"!F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 207 "de := diff(q(t),t$2)+(q(t)^2-1)*diff(q(t),t)+q(t)=0; \nic := q(0)=1,D(q)(0)=0;\nQn := desolve(\{de,ic\},q(t),t=0..40,type=n umeric,method=rk78);\nplot('Qn'(t),t=0..30,color=magenta,thickness=2,l abels=[`t`,`q(t)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%di ffG6$-%\"qG6#%\"tG-%\"$G6$F-\"\"#\"\"\"*&,&*$)F*F1F2F2F2!\"\"F2-F(6$F* F-F2F2F*F2\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"qG6#\" \"!\"\"\"/--%\"DG6#F(F)F*" }}{PARA 13 "" 1 "" {GLPLOT2D 709 253 253 {PLOTDATA 2 "6'-%'CURVESG6#7cx7$$\"\"!F)$\"\"\"F)7$$\"+5G$R<)!#6$\"+'= 5m'**!#57$$\"+ilyM;F2$\"+uHhm)*F27$$\"+W)z@X#F2$\"+$32/q*F27$$\"+DJdpK F2$\"+Z[To%*F27$$\"+)ofV!\\F2$\"+`rw2))F27$$\"+]i9RlF2$\"+j@*=)yF27$$ \"+XV)RQ*F2$\"+msX&f&F27$$\"+WA)GA\"!\"*$\"+w_m5BF27$$\"+U\"[GQ\"FT$!+ 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T7$$\"+b!)[/HF^dl$!+nXq2?FT7$$\"+=,27HF^dl$!+^Ke1?FT7$$\"+!=_'>HF^dl$! +#e%Q&*>FT7$$\"+VUBFHF^dl$!+d.=w>FT7$$\"+1j\"[$HF^dl$!+u$p0&>FT7$$\"+` \"3u'HF^dl$!+CIj!z\"FT7$$\"#IF)$!+)\\&fu:FT-%*THICKNESSG6#\"\"#-%'COLO URG6&%$RGBG$\"*++++\"F^dlF(Fa_o-%+AXESLABELSG6$%\"tG%%q(t)G-%%VIEWG6$; F(Fe^o%(DEFAULTG" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "The solution (charge " }{TEXT 276 1 "q" }{TEXT -1 141 ") \+ appears to be approach a periodic (steady-state) waveform. The differe nce between successive zeros tends to the constant value 3.331643430. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "zrs := [evalf[12](allbisect(Qn,0..39,sensitivity=4))];\nnops(z rs);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$zrsG7.$\"-h+(G#z8!#6$\"-mNK )>g%F($\"-VvjdEzF($\"-!o\\+e7\"!#5$\"-74U'*e9F/$\"-fL&G@z\"F/$\"--wGHD @F/$\"-(*=sXeCF/$\"-$>c@;z#F/$\"-!\\!fyCJF/$\"-(yC]zX$F/$\"-$3f96z$F/ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "for i to 11 do\n ev alf(evalf[12](zrs[i+1]-zrs[i]));\nend do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+NXvAK!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+SJ fCL!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+$fG9L$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+BrjJL!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$\"+XKkJL!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+CMkJL!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+IMkJL!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+IMkJL!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+IM kJL!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+IMkJL!\"*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+IMkJL!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "The period of the steady-state solution is therefore approximately 2 " }{TEXT 267 1 "x" }{TEXT -1 28 " 3.3316 43430 = 6.663286860." }}{PARA 0 "" 0 "" {TEXT -1 71 "The differences \+ between alternate zeros approach the value 6.663286859." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "for i t o 10 by 2 do\n evalf(evalf[12](zrs[i+2]-zrs[i]));\nend do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+vwMZl!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+;d1jm!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+pmGjm!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+foGjm!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+foGjm!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 60 "We can also construct a numerical procedu re for the current " }{XPPEDIT 18 0 "i = ``;" "6#/%\"iG%!G" }{XPPEDIT 18 0 "dq/dt = `q '`(t);" "6#/*&%#dqG\"\"\"%#dtG!\"\"-%$q~'G6#%\"tG" } {TEXT -1 21 " and plot its graph. 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" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 215 "aa : = 31.2478590491;\nbb := 37.9111459085;\nT := bb-aa;\nQn_sstate := t -> Qn(t-T*floor((t-aa)/T));\nplot('Qn_sstate'(t),t=0..30,color=red,thick ness=2,\n title=\"steady state part of solution\",labels=[`t`,` q(t)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#aaG$\"-\"\\!fyCJ!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bbG$\"-&3f96z$!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG$\"*'oGjm!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*Qn_sstateGf*6#%\"tG6\"6$%)operatorG%&arrowGF(-%#QnG6#,&9$\"\"\"* &%\"TGF1-%&floorG6#*&,&F0F1%#aaG!\"\"F1F3F:F1F:F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 644 218 218 {PLOTDATA 2 "6(-%'CURVESG6#7]x7$$\"\"!F)$\" +P8y&e\"!\"*7$$\"+DJdpK!#5$\"+zty=8F,7$$\"+]i9RlF0$\"+BPA4)*F07$$\"++` chzF0$\"+3NaB!)F07$$\"+XV)RQ*F0$\"+['zz(fF07$$\"+R.k!3\"F,$\"+(Q_.h$F0 7$$\"+WA)GA\"F,$\"+X60c&)!#67$$\"+$>lGI\"F,$!+4\"f&=))FL7$$\"+U\"[GQ\" F,$!+&Rxgv#F07$$\"+#4JGY\"F,$!+u&)faZF07$$\"+TS\"Ga\"F,$!+M_)z%oF07$$ \"+!*pzA;F,$!+U#*3')*)F07$$\"+S*zFq\"F,$!+#*\\%)46F,7$$\"+*)Gw#y\"F,$! +1C958F,7$$\"+Qeui=F,$!+:W?\"\\\"F,7$$\"+%4sP-#F,$!+'o0Ox\"F,7$$\"+]$) z%=#F,$!+NL%p$>F,7$$\"+y9JlAF,$!+*4p)z>F,7$$\"+1Y#eM#F,$!+G6T-?F,7$$\" +MxLECF,$!+thf3?F,7$$\"+i3&o]#F,$!+'H%*=+#F,7$$\"+;FPFGF,$!+%pGR*=F,7$ $\"+pX*y9$F,$!+Z^O7fS*\\F,$\"++\"oS+\"F,7$$\"+D#fE:&F,$\" +'\\+'*R\"F,7$$\"+b#f7J&F,$\"+#\\!=3F,7$$\"+ >$f%GcF,$\"+u_)3*>F,7$$\"+I\"H+r&F,$\"+v?n1?F,7$$\"+X*)f\"z&F,$\"+%o)= 2?F,7$$\"+g(oJ(eF,$\"+E.w&*>F,7$$\"+q&QZ&fF,$\"+bu+v>F,7$$\"++#yy6'F,$ \"+a_%H\">F,7$$\"+Dy,\"G'F,$\"+#[Z7$=F,7$$\"+qZSolF,$\"+t$*4`;F,7$$\"+ 7F,7$$\"+!pL3#*)F,$!+d[ow >F,7$$\"+g)pj&*)F,$!+hXt*)>F,7$$\"+Dg!>**)F,$!+D\")4**>F,7$$\"+!>Uu-*F ,$!+.A70?F,7$$\"+g$yH1*F,$!+-H83?F,7$$\"+!p]S8*F,$!+cqH1?F,7$$\"+DI70# *F,$!+&\\(p&*>F,7$$\"+g`>w#*F,$!+E`.y>F,7$$\"+%pnsM*F,$!+=]ka>F,7$$\"+ 0_J&o*F,$!+*o\"***y\"F,7$$\"+siL-5!\")$!+[(zZc\"F,7$$\"+JL(4.\"F_bl$!+ wn!)H8F,7$$\"+!R5'f5F_bl$!+\\&)fT5F,7$$\"+WiEw5F_bl$!+y)R$*R)F07$$\"+( 4AH4\"F_bl$!+D*\\7/'F07$$\"+]zd46F_bl$!+))*yRC$F07$$\"+/QBE6F_bl$\"+WJ 5\\%*!#77$$\"+dq(49\"F_bl$\"+7,wTNF07$$\"+5.sb6F_bl$\"++fTmtF07$$\"+iN Yq6F_bl$\"+([Wx7\"F,7$$\"+:o?&=\"F_bl$\"+w!*Gz9F,7$$\"++DQ,7F_bl$\"+2R $ow\"F,7$$\"+%=ev@\"F_bl$\"+y;5M>F,7$$\"+EgkD7F_bl$\"+F,7$$\"+pQt L7F_bl$\"+hP\"=+#F,7$$\"+7<#=C\"F_bl$\"+\\lh3?F,7$$\"+a&4*\\7F_bl$\"+( =%H-?F,7$$\"+3dr!G\"F_bl$\"+Yb1,>F,7$$\"+j=_68F_bl$\"+3F.I7F,7$$\"+%\\m0R\"F_bl$\"+ *G4i1\"F,7$$\"+V^K09F_bl$\"+t#pZ#*)F07$$\"+#z$3?9F_bl$\"+AZAOpF07$$\"+ UC%[V\"F_bl$\"+sPCKYF07$$\"+R;w]9F_bl$\"+%4s%4&)\\\"F_bl$!+c:9S**F07$$\"+3 7F,7$$\"+.[0::F_bl$!+/5J19F,7$$\"+*eAL_\"F_bl$!+Hx#*z:F,7$$\"+v.fJ:F_b l$!+q#pLs\"F,7$$\"+Zf7[:F_bl$!+vpy9>F,7$$\"+>:mk:F_bl$!+Ol:(*>F,7$$\"+ _'e=d\"F_bl$!+h>s2?F,7$$\"+%yb!z:F_bl$!+u.)o+#F,7$$\"+;HD'e\"F_bl$!+&H jo*>F,7$$\"+[+X$f\"F_bl$!+`OZz>F,7$$\"+7V%yg\"F_bl$!+R!)*y#>F,7$$\"+w& QAi\"F_bl$!+w([)f=F,7$$\"+v4L`;F_bl$!+L1`t;F,7$$\"+uLU%o\"F_bl$!+\\s/T 9F,7$$\"+k[a;=F_bl$\"+P9?'e'F07$$\"+U!=n#=F_bl$ \"+\"HHfh)F07$$\"+u#>V$=F_bl$\"+eYYj5F,7$$\"+10#>%=F_bl$\"+fr=d7F,7$$ \"+qH7d=F_bl$\"+8J3#f\"F,7$$\"+MaKs=F_bl$\"+k#Gk#=F,7$$\"+1`w!)=F_bl$ \"+Rbb5>F,7$$\"+y^?*)=F_bl$\"+q#[d'>F,7$$\"+9^U$*=F_bl$\"+td&R)>F,7$$ \"+]]k(*=F_bl$\"+&*>s'*>F,7$$\"+')\\'=!>F_bl$\"+!*yj/?F,7$$\"+A\\31>F_ bl$\"+F_bl$\"+2md/?F,7$$\"+mY'H#>F_bl$\"+(4<\"*)>F ,7$$\"+QXSJ>F_bl$\"+!fBX'>F,7$$\"+6W%)R>F_bl$\"+l\"eF$>F,7$$\"+8)y,(>F _bl$\"+V]Cx$F07$$\"+4w)R7#F_bl$\"+$Rlu\"GFL7 $$\"+#44?8#F_bl$!+T\\.3:F07$$\"+w0.S@F_bl$!+nGoKMF07$$\"+g?0[@F_bl$!++ 1IuaF07$$\"+WN2c@F_bl$!+IfB'f(F07$$\"+G]4k@F_bl$!+jSRS(*F07$$\"+6l6s@F _bl$!+@7)H=\"F,7$$\"+%*z8!=#F_bl$!+i2&yP\"F,7$$\"+y%f\")=#F_bl$!+FKr]: F,7$$\"+gYD.AF_bl$!+@Y!))z\"F,7$$\"+T)\\$=AF_bl$!+*G$yU>F,7$$\"+Ku*eA# F_bl$!+o!f8)>F,7$$\"+A]WLAF_bl$!+#e)>-?F,7$$\"+8E*4C#F_bl$!+zyh3?F,7$$ \"+/-a[AF_bl$!+@<].?F,7$$\"+Ly4!G#F_bl$!+Q?e,>F,7$$\"+ial6BF_bl$!+rZ'f s\"F,7$$\"+7)3DM#F_bl$!+P=+DF_bl$\"+B'[70\"F,7$$\"+u))3;DF_bl$ \"+I^uS9F,7$$\"+ed*>`#F_bl$\"+'oDst\"F,7$$\"+UE!za#F_bl$\"+.q-<>F,7$$ \"+E&4Qc#F_bl$\"+X'>j*>F,7$$\"+5rprDF_bl$\"+kE#z+#F,7$$\"+$p%ezDF_bl$ \"+up&f+#F,7$$\"+wAZ(e#F_bl$\"+n\\I$*>F,7$$\"+g)f`f#F_bl$\"+!pPA(>F,7$ $\"+F]86EF_bl$\"+sJK6>F,7$$\"+%>5pi#F_bl$\"+d:AK=F,7$$\"+u;!fl#F_bl$\" +TFe_;F,7$$\"+bJ*[o#F_bl$\"+Eq%HV\"F,7$$\"+4p],FF_bl$\"+h2b'G\"F,7$$\" +j17=FF_bl$\"+))3S@6F,7$$\"+a\")o>F,7$$\"+b!)[/HF_bl$!+nXq2?F,7$$\"+=,27HF_bl$!+^ Ke1?F,7$$\"+!=_'>HF_bl$!+#e%Q&*>F,7$$\"+VUBFHF_bl$!+d.=w>F,7$$\"+1j\"[ $HF_bl$!+u$p0&>F,7$$\"+`\"3u'HF_bl$!+CIj!z\"F,7$$\"#IF)$!+)\\&fu:F,-%& TITLEG6#Q>steady~state~part~of~solution6\"-%*THICKNESSG6#\"\"#-%'COLOU RG6&%$RGBG$\"*++++\"F_blF(F(-%+AXESLABELSG6$%\"tG%%q(t)G-%%VIEWG6$;F(F g\\o%(DEFAULTG" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "It is now possible to obtain a numerical procedure for the " } {TEXT 257 14 "transient part" }{TEXT -1 18 " of the solution. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "Qn_trans := Qn-Qn_sstate:\nplot('Qn_trans'(t),t=0..10,q_trans,col or=COLOR(RGB,0,.7,0),\n thickness=2,labels=[`t`,`q(t)`],title=\"t ransient part of solution\");" }}{PARA 13 "" 1 "" {GLPLOT2D 660 273 273 {PLOTDATA 2 "6(-%'CURVESG6#7ao7$$\"\"!F)$!*P8y&e!\"*7$$\"+3x&)*3\" !#5$!+&H#f!4&F07$$\"+;arz@F0$!+e,N!Q%F07$$\"+!y%*z7$F0$!+5$[G!QF07$$\" +XTFwSF0$!++cRcKF07$$\"+qMrU^F0$!+(z.)pEF07$$\"+\"z_\"4iF0$!+np[,@F07$ $\"+l6m#G(F0$!+EtiM:F07$$\"+S&phN)F0$!*79De*F07$$\"+:ddC%*F0$!*Q>4f$F0 7$$\"+*=)H\\5F,$\"*v(\\CGF07$$\"+=JN[6F,$\"*6eBH*F07$$\"+[!3uC\"F,$\"+ [6&ej\"F07$$\"+!pt*\\8F,$\"+n6UJCF07$$\"+J$RDX\"F,$\"+322uKF07$$\"+kGh e:F,$\"+iVyUTF07$$\"+)R'ok;F,$\"+3x*H\"\\F07$$\"+_(>/x\"F,$\"+-F!pX&F0 7$$\"+1J:w=F,$\"+MTdlcF07$$\"+WIM.>F,$\"*'4\"yl&F,7$$\"+#)H`I>F,$\"*Xl \\i&F,7$$\"+?Hsd>F,$\"*F`yc&F,7$$\"+dG\"\\)>F,$\"*XPF,7$$\"+LY.KNF,$\")> 1RDF,7$$\"+\"o7Tv$F,$\")vZ`:F,7$$\"+$Q*o]RF,$\"(;\\a)F,7$$\"+\"=lj;%F, $\"),)pR\"F07$$\"+V&RY2aF,$!)E)>U#F,7$$\"+yXu9cF,$!)gM\"\\\"F,7$$\"+\\y))GeF,$!(oNP)F, 7$$\"+i_QQgF,$!(%H!z%F,7$$\"+!y%3TiF,$!(0-!HF,7$$\"+O![hY'F,$!(*=ZF07$ $\"+2s?6zF,$\")yi!f%F[x7$$\"+IXaE\")F,$\"(8)\\xF07$$\"+l*RRL)F,$\"(K<+ \"F,7$$\"+`<.Y&)F,$\"'>#o*F,7$$\"+8tOc()F,$\"'j()pF,7$$\"+\\Qk\\*)F,$ \"'`&Q%F,7$$\"+p0;r\"*F,$\"'X+CF,7$$\"+lxGp$*F,$\"'z79F,7$$\"+!oK0e*F, $\"&*o$)F,7$$\"+<5s#y*F,$\"&2I&F,7$$\"#5F)$\"&TO$F,-%&COLORG6&%$RGBGF( $\"\"(!\"\"F(-%&TITLEG6#Q;transient~part~of~solution6\"-%*THICKNESSG6# \"\"#-%+AXESLABELSG6$%\"tG%%q(t)G-%%VIEWG6$;F(Fi`l%(DEFAULTG" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "The graphs of th e transient and steady state parts of the solution can be plotted alon g with the complete solution. 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "plot('In(t)',t=30..40,color=coral,thickne ss=2,labels=[`t`,`i(t)`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 523 268 268 {PLOTDATA 2 "6'-%'CURVESG6#7]p7$$\"#I\"\"!$\"+^s<\"R(!#57$$\"+arz@I!\" )$\"+%oZB`)F-7$$\"+TFwSIF1$\"++(HSw*F-7$$\"+G:4iIF1$\"+*>'ye6!\"*7$$\" +&phN3$F1$\"+/_.89F>7$$\"+>)H\\5$F1$\"+J7oi7$$\"+03uCJF1$\"+,Qtr@F> 7$$\"+pt*\\8$F1$\"+T)e)*Q#F>7$$\"+LRDXJF1$\"+nh^tDF>7$$\"+5wb]JF1$\"+m &Q\"REF>7$$\"+'Ghe:$F1$\"+EY[uEF>7$$\"+CJ^eJF1$\"+zxPyEF>7$$\"+j\\;hJF 1$\"+E&H>n#F>7$$\"+-o\"Q;$F1$\"+]0VaEF>7$$\"+S'ok;$F1$\"+WbMDEF>7$$\"+ w>/xJF1$\"+4lU\"R#F>7$$\"+6`h(=$F1$\"+3.o#*>F>7$$\"+'G\"\\)>$F1$\"+Y$3 \"*[\"F>7$$\"+hsO4KF1$\"+4Hox)*F-7$$\"+Eo%*=KF1$\"+(z!)R.'F-7$$\"+!RE& GKF1$\"+qIVBHF-7$$\"+62JRKF1$\"+`,G&p#!#67$$\"+K]4]KF1$!+9-\">l\"F-7$$ \"+QAvrKF1$!+-_PnSF-7$$\"+oHi#H$F1$!+D$y\\W&F-7$$\"+!fv:J$F1$!+G7M&Q'F -7$$\"+\"47TL$F1$!+eW!pV(F-7$$\"+jM?`LF1$!+'zK:V)F-7$$\"+o7TvLF1$!+Mnp t)*F-7$$\"+Q*o]R$F1$!+@P#p:\"F>7$$\"+=lj;MF1$!+*)*)y69F>7$$\"+aR7$$\"+9EgeMF1$!+!o')o=#F>7$$\"+9_&R#F>7$$\"+93GyMF 1$!+X[[rDF>7$$\"+8se$[$F1$!+&Q7yj#F>7$$\"+7O*))[$F1$!+t^.uEF>7$$\"+7oa \"\\$F1$!+L7$$\"+6+?%\\$F1$!+f]\\sEF>7$$\"+5K&o\\$F1$!+Rs`bEF>7$ $\"+5k]*\\$F1$!+vF+FEF>7$$\"+D,`5NF1$!+#y%R\"Q#F>7$$\"+SQb@NF1$!+fd!)e >F>7$$\"+;+:JNF1$!+gs;8:F>7$$\"+#>Y2a$F1$!+)y%\\m5F>7$$\"+D.6^NF1$!+:9 O-kF-7$$\"+eWZhNF1$!+@PY&)HF-7$$\"+A;=sNF1$!+k*R8J$Fcr7$$\"+&y))Ge$F1$ \"+!4Pef\"F-7$$\"+E&QQg$F1$\"+MZ'z(RF-7$$\"+y%3Ti$F1$\"+BaJ_`F-7$$\"+/ [hYOF1$\"+I&)RskF-7$$\"+Qx$om$F1$\"+Xrq:uF-7$$\"+P+V)o$F1$\"+(o')*\\&) F-7$$\"+qe*zq$F1$\"+'Q%fI)*F-7$$\"+#\\'QHPF1$\"+0\"*\\o6F>7$$\"+,M^\\P F1$\"+Y))z29F>7$$\"+?=bqPF1$\"+:!>%\\7$$\"+@27\"z$F1$\"+%)o%G<#F>7$ $\"+(e()=!QF1$\"+T->,CF>7$$\"+`Wl7QF1$\"+R7$))e#F>7$$\"+'*oC:QF1$\"+1O r@EF>7$$\"+Q$Ry\"QF1$\"+?H&zk#F>7$$\"+\"yJ/#QF1$\"+A0hmEF>7$$\"+CU-BQF 1$\"+![vnn#F>7$$\"+nmhDQF1$\"+7vfxEF>7$$\"+5\"4#GQF1$\"+[EKoEF>7$$\"+` :!3$QF1$\"+K*H$[EF>7$$\"+'*RRLQF1$\"+jo;7$$\"+'e)*R%QF1$\"+$GH=P#F >7$$\"+vJgaQF1$\"+!fnQ'>F>7$$\"+`*>^'QF1$\"+Q)Q[Z\"F>7$$\"+JnjvQF1$\"+ 8N[.**F-7$$\"+e0I&)QF1$\"+;o'[-'F-7$$\"+&Qk\\*QF1$\"+^:A#*GF-7$$\"+@-/ 1RF1$\"+ZV_i=Fcr7$$\"+dg6 " 0 "" {MPLTEXT 1 0 101 "evalf[12](findmax ('In'(t),t=31.5));\nevalf[12](findmax('In'(t),t=38.2));\nevalf(evalf[1 2](%[1]-%%[1]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"-a3\"=#eJ!#5$ \"-hz9WyE!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"-[%zYX#Q!#5$\"-gz 9WyE!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+foGjm!\"*" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Taking " }{XPPEDIT 18 0 "a = 31.5821810854;" "6#/%\"aG-%&FloatG6$\"-a3\"=#eJ!#5" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b = 38.2454679448;" "6#/%\"bG-%&FloatG6$ \"-[%zYX#Q!#5" }{TEXT -1 103 " in the formula (i), we can set up proce dures for the transient and steady state parts of the current. 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"Example 2: " }{XPPEDIT 18 0 "mu= 2" "6#/%#muG\"\"#" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 59 "We find a numerical solution for the differential equation " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*q/(d*t^2)+2*(q^2-1);" "6#,&*(%\"dG\"\"# %\"qG\"\"\"*&F%F(*$%\"tGF&F(!\"\"F(*&F&F(,&*$F'F&F(F(F,F(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dq/dt+q = 0;" "6#/,&*&%#dqG\"\"\"%#dtG!\"\"F' %\"qGF'\"\"!" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 28 "with the \+ initial conditions " }{XPPEDIT 18 0 "q(0)=1" "6#/-%\"qG6#\"\"!\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "`q '`(0) = 0" "6#/-%$q~'G6#\"\"!F'" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 298 "de := diff(q(t),t$2)+2*(q(t)^2-1)*diff(q(t),t)+ q(t)=0;\nic := q(0)=1,D(q)(0)=0;\n(Qn,In) := desolve(\{de,ic\},q(t),t= 0..25,type=numeric,method=rk78,output=proc_pair);\nplot('Qn(t)',t=0..2 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{TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*q/(d*t^2) +5*(q^2-1);" "6#,&*(%\"dG\"\"#%\"qG\"\"\"*&F%F(*$%\"tGF&F(!\"\"F(*&\" \"&F(,&*$F'F&F(F(F,F(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dq/dt+q = 0; " "6#/,&*&%#dqG\"\"\"%#dtG!\"\"F'%\"qGF'\"\"!" }{TEXT -1 1 "," }} {PARA 0 "" 0 "" {TEXT -1 28 "with the initial conditions " }{XPPEDIT 18 0 "q(0)=1" "6#/-%\"qG6#\"\"!\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "`q '`(0) = 0" "6#/-%$q~'G6#\"\"!F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 298 "de := di ff(q(t),t$2)+5*(q(t)^2-1)*diff(q(t),t)+q(t)=0;\nic := q(0)=1,D(q)(0)=0 ;\n(Qn,In) := desolve(\{de,ic\},q(t),t=0..35,type=numeric,method=rk78, output=proc_pair);\nplot('Qn(t)',t=0..35,thickness=2,color=magenta,lab els=[`t`,`q(t)`]);\nplot('In(t)',t=0..35,thickness=2,color=coral,label s=[`t`,`i(t)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6 $-%\"qG6#%\"tG-%\"$G6$F-\"\"#\"\"\"*(\"\"&F2,&*$)F*F1F2F2F2!\"\"F2-F(6 $F*F-F2F2F*F2\"\"!" }}{PARA 11 "" 1 "" 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" }}{PARA 0 "" 0 "" {TEXT -1 65 "Specifically, construct numerical procedures for both the charge \+ " }{TEXT 283 1 "q" }{TEXT -1 17 " and the current " }{XPPEDIT 18 0 "i= dq/dt" "6#/%\"iG*&%#dqG\"\"\"%#dtG!\"\"" }{TEXT -1 19 " over the inter val " }{XPPEDIT 18 0 "0<=t" "6#1\"\"!%\"tG" }{XPPEDIT 18 0 "``<=15" "6 #1%!G\"#:" }{TEXT -1 109 " and plot the graphs of the charge and the c urrent over this interval. Also draw the associated phase curve. " }} {PARA 0 "" 0 "" {TEXT -1 77 "Find the maximum magnitude of the current in the circuit during the interval " }{XPPEDIT 18 0 "0<=t" "6#1\"\"!% \"tG" }{XPPEDIT 18 0 "``<=15" "6#1%!G\"#:" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 30 "______________________________" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 30 "__ ____________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 54 "Investigate the response of a series LCR circuit with " }{XPPEDIT 18 0 "L=1" "6#/ %\"LG\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "R=1/4" "6#/%\"RG*&\"\"\" F&\"\"%!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "C=1" "6#/%\"CG\"\"\"" } {TEXT -1 21 " to an input voltage " }{XPPEDIT 18 0 "V(t) = sin*3*t/(t^ 2+32);" "6#/-%\"VG6#%\"tG**%$sinG\"\"\"\"\"$F*F'F*,&*$F'\"\"#F*\"#KF*! \"\"" }{TEXT -1 24 ", given that the charge " }{TEXT 280 1 "q" }{TEXT -1 34 " on the capacitor and the current " }{XPPEDIT 18 0 "i = dq/dt; " "6#/%\"iG*&%#dqG\"\"\"%#dtG!\"\"" }{TEXT -1 26 " are intially equal \+ to 0. 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7$$\"+oqA]&*Fcr$\"+!Hd.$=FB7$$\"+'*R&Gn*Fcr$\"+OE+ZPFB7$$\"+5LWK)*Fcr$ \"+7oE9bFB7$$\"+Z!HD+\"F^s$\"+sXphqFB7$$\"+2ljC5F^s$\"+`[fF$)FB7$$\"+N 6&*[5F^s$\"+2-]h#*FB7$$\"+h$pZ2\"F^s$\"+u(yh#)*FB7$$\"+vwK,6F^s$\"+,'= \"****FB7$$\"+fI$y7\"F^s$\"+C^Ut(*FB7$$\"+/Y\\`6F^s$\"+Hg4e\"*FB7$$\"+ []bx6F^s$\"+XDmx\")FB7$$\"+`4K*>\"F^s$\"+i9@roFB7$$\"+g.>=7F^s$\"+ho#3 H&FB7$$\"+(ywOB\"F^s$\"+!p8&*\\$FB7$$\"+:'GaC\"F^s$\"+]foo:FB7$$\"+;JC `7F^s$!+sM!oC%Fgal-%*AXESSTYLEG6#%%NONEG" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 83 "Note that the differential equation which governs the behaviour of the circuit is: " }}{PARA 256 "" 0 "" {TEXT -1 3 " \+ " }{XPPEDIT 18 0 "d^2*q/(d*t^2)+1/4;" "6#,&*(%\"dG\"\"#%\"qG\"\"\"*&F %F(*$%\"tGF&F(!\"\"F(*&F(F(\"\"%F,F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 " dq/dt+q = sin(3*t)/(t^2+32)" "6#/,&*&%#dqG\"\"\"%#dtG!\"\"F'%\"qGF'*&- %$sinG6#*&\"\"$F'%\"tGF'F',&*$F1\"\"#F'\"#KF'F)" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 44 "and it is subject to the initial conditio ns " }{XPPEDIT 18 0 "q(0) = 0;" "6#/-%\"qG6#\"\"!F'" }{TEXT -1 4 ", q' " }{XPPEDIT 18 0 "``(0)=0" "6#/-%!G6#\"\"!F'" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 65 "Specifically, construct numerical procedu res for both the charge " }{TEXT 281 1 "q" }{TEXT -1 17 " and the curr ent " }{XPPEDIT 18 0 "i=dq/dt" "6#/%\"iG*&%#dqG\"\"\"%#dtG!\"\"" } {TEXT -1 19 " over the interval " }{XPPEDIT 18 0 "0<=t" "6#1\"\"!%\"tG " }{XPPEDIT 18 0 "``<=15" "6#1%!G\"#:" }{TEXT -1 109 " and plot the gr aphs of the charge and the current over this interval. Also draw the a ssociated phase curve. " }}{PARA 0 "" 0 "" {TEXT -1 77 "Find the maxim um magnitude of the current in the circuit during the interval " } {XPPEDIT 18 0 "0<=t" "6#1\"\"!%\"tG" }{XPPEDIT 18 0 "``<=15" "6#1%!G\" #:" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 30 "__________________ ____________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 30 "______________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "Q3 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 33 " Given the differential equation " }} {PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*q/(d*t^2)+2*(q^ 2-1);" "6#,&*(%\"dG\"\"#%\"qG\"\"\"*&F%F(*$%\"tGF&F(!\"\"F(*&F&F(,&*$F 'F&F(F(F,F(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dq/dt+q = 0;" "6#/,&*&% #dqG\"\"\"%#dtG!\"\"F'%\"qGF'\"\"!" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 28 "with the initial conditions " }{XPPEDIT 18 0 "q(0)=1" "6# /-%\"qG6#\"\"!\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "`q '`(0) = 0" " 6#/-%$q~'G6#\"\"!F'" }{TEXT -1 53 ", construct numerical procedures fo r both the charge " }{TEXT 274 1 "q" }{TEXT -1 17 " and the current " }{XPPEDIT 18 0 "i = dq/dt;" "6#/%\"iG*&%#dqG\"\"\"%#dtG!\"\"" }{TEXT -1 19 " over the interval " }{XPPEDIT 18 0 "0<=t" "6#1\"\"!%\"tG" } {XPPEDIT 18 0 "``<=50" "6#1%!G\"#]" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 19 "Assuming that both " }{TEXT 270 1 "q" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "i = dq/dt;" "6#/%\"iG*&%#dqG\"\"\"%#dtG!\"\"" } {TEXT -1 30 " are essentially periodic for " }{XPPEDIT 18 0 "t>=30" "6 #1\"#I%\"tG" }{TEXT -1 58 ", find the period of the steady state (peri odic) parts of " }{TEXT 271 1 "q" }{TEXT -1 5 " and " }{TEXT 272 1 "i " }{TEXT -1 17 " as functions of " }{TEXT 273 1 "t" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 81 "Plot graphs which illustrate the decompo sition of both the curves for the charge " }{TEXT 277 1 "q" }{TEXT -1 17 " and the current " }{TEXT 278 1 "i" }{TEXT -1 46 " into their tran sient and steady state parts. " }}{PARA 0 "" 0 "" {TEXT -1 30 "_______ _______________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 30 "______________________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 25 "Code for drawing pictures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 32 "Code for drawing the LCR circuit" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 716 "coil := PLOT(CURVES([evalf(seq([-cos(r/5)+r/15,si n(r/5)],r=0..173))])):\nsource :=\nPLOT(CURVES([evalf(seq([3*cos(r/5)- 12.5,3*sin(r/5)-10],r=0..32))])):\nwire := PLOT(CURVES([[12.54,0],[24, 0]],[[-12.5,0],[-1,0]],\n[[24,0],[24,-9.4]],[[24,-10.6],[24,-20]],[[20 ,-9.4],[28,-9.4]],\n[[20,-10.6],[28,-10.6]],[[-12.5,0],[-12.5,-7]],\n[ [-12.5,-13],[-12.5,-20]],[[-12.5,-20],[-0.5,-20]],\n[[12.5,-20],[24,-2 0]])):\nresistor := PLOT(CURVES([[-0.5,-20],[0,-18.5],[1,-21.5],[2,-18 .5],\n[3,-21.5],[4,-18.5],[5,-21.5],[6,-18.5],[7,-21.5],[8,-18.5],\n[9 ,-21.5],[10,-18.5],[11,-21.5],[12,-18.5],[12.5,-20]])):\nt1:=textplot( [[7,4,`L`],[27.5,-6.5,`C`],[7,-23,`R`],[-12.5,-10,`V(t)`]]):\nplots[di splay](\{coil,wire,source,resistor,t1\},axes=NONE);" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "5 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }